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and the line are very different objects. One cannot stretch the circle to look like the line, nor compress the line to fit on the circle without gaps or overlaps. However, a small piece of the circle can be stretched and flattened out to look like a small piece of the line. For this reason, one
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Here, note that condition (2) is for the most part stronger than condition (1), and that extra caution should be taken to distinguish between the two. For example, some variation in the definition of
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of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain.
247:. In which case, a property is said to be local if it can be detected from the local subgroups. Global and local properties formed a significant portion of the early work on the
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if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is
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of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.
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and the plane are locally equivalent. A small enough observer standing on the
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Perhaps the best-known example of the idea of locality lies in the concept of
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can arise as a result of the different choices of these conditions.
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make it natural to take a "small neighborhood" of a ring to be the
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may say that the circle and the line are locally equivalent.
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over a commutative ring is a local property, but being a
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327:"Definition of local-maximum | Dictionary.com"
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251:, which was carried out during the 1960s.
187:, a "small neighborhood" is taken to be a
201:if every finitely generated subgroup is
140:Given some notion of equivalence (e.g.,
67:is sometimes said to exhibit a property
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249:classification of finite simple groups
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376:"Maxima, minima, and saddle points"
194:. An infinite group is said to be
39:Properties of a point on a function
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265:For commutative rings, ideas of
86:of sets exhibiting the property.
255:Properties of commutative rings
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136:Properties of a pair of spaces
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179:Properties of infinite groups
205:. For instance, a group is
59:Properties of a single space
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217:Properties of finite groups
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281:. For instance, being a
303:Local path connectedness
291:Localization of a module
79:exhibiting the property;
308:Local-global principle
289:is not. For more, see
114:Locally path-connected
356:mathworld.wolfram.com
350:Weisstein, Eric W.
122:, Locally regular,
331:www.dictionary.com
267:algebraic geometry
240:of the nontrivial
189:finitely generated
159:For instance, the
154:topological spaces
130:Locally metrizable
116:topological spaces
106:topological spaces
26:sufficiently small
120:Locally Hausdorff
110:Locally connected
84:neighborhood base
82:Each point has a
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30:arbitrarily small
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383:. Retrieved
380:Khan Academy
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359:. Retrieved
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35:of points).
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287:free module
283:flat module
279:local rings
275:prime ideal
238:normalizers
18:mathematics
399:Categories
385:2019-11-30
361:2019-11-30
336:2019-11-30
314:References
261:local ring
245:-subgroups
152:) between
297:See also
196:locally
192:subgroup
150:isometry
98:Examples
211:soluble
183:For an
173:surface
69:locally
22:locally
236:, the
169:sphere
161:circle
126:etc...
273:at a
221:For
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